Monday, May 21, 2012
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We often say that the primary reason why string/M-theory is so essential for modern physics is that it is the only known – and most likely, the only mathematically possible – consistent theory of gravity. Everyone who believes that he or she can do state-of-the-art research of quantum gravity without string theory is an unhinged crank, a barbarian, and a conspiracy theorist of the same kind as those who believe that Elvis Presley lives on the Moon.

But another reason why string/M-theory is indispensable for the 21st century theoretical and particle physics is that many of the "ordinary", important, non-gravitational quantum field theories and some of their non-field-theoretical but still non-gravitational generalizations are tightly embedded as limits in string theory. In this way, a theory whose main strength is to provide us with robust quantum rules governing gravity is important for our knowledge of contexts that avoid gravity, too.

Because of the dense network of relationships within string theory that link ideas, concepts, and equations that used to be considered independent – and I mostly mean dualities but not only dualities – each of the "ordinary" non-gravitational theories may be analyzed from new perspectives. In particular, extreme limits of the old theories in which a quantity is sent to infinity (or zero) could have been very mysterious but many of the mysteries go away as string/M-theory allows us to use new descriptions.

Among the new insights that we're learning from the stringy network of ideas, rules, equations, and maps, we also encounter new quantum field theories – and some other non-gravitational generalizations of these theories which are not quantum field theories – i.e. theories that are not full-fledged string vacua and that we shouldn't have overlooked in the past but we have. What are they?

In March, I discussed the maximally supersymmetric gauge theory in four dimensions. It's arguably the most far-reaching or at least the most widely studied example of the point I made in the second paragraph.

The \(\NNN=4\) gauge theory in \(d=4\) is a gauge theory with 16 real supercharges. If you write it in terms of components, it's a gauge theory with a gauge group – it can be \(SU(N)\), \(O(N)\), \(OSp(2k)\), \(E_6\), or any other compact Lie group – which is coupled to four Weyl neutrinos in the adjoint representation of the same group and six Hermitian scalars in the same representation. When the interactions are appropriately chosen, we discover that the theory has those 16 supersymmetries even at the interacting level.

Nima Arkani-Hamed would call this theory a harmonic oscillator of the 21st century. Andy Strominger reserves this term for black holes but it's true that these two theoretical constructs are perhaps even more important if they work as a team and they often do.

String theory tells us lots of things about the seemingly ordinary gauge theory which wasn't known to have any direct connection to strings. In fact, we have known for almost 15 years that this gauge theory is string theory. The \(SU(N)\) maximally supersymmetric gauge theory is totally equivalent to the superselection sector of type IIB string theory respecting the asymptotic conditions of \(AdS_5\times S^5\). This relationship is, of course, the most famous example of Juan Maldacena's AdS/CFT correspondence.

However, the remarkable relationship was found – and may be "almost proven" – by less shocking relationships between this gauge theory and string theory. In particular, the simplest representation of the gauge theory is the dynamics of D3-branes in type IIB string theory at very long distances. Some properties of the gauge theory may be deduced out of this realization immediately. In particular, the theory inherits the \(SL(2,\ZZ)\) S-duality group – which includes the \(g\to 1/g\) exchange of the weak coupling with the strong coupling – from the full type IIB string theory. In the type IIB string theory, the S-duality group may also be motivated by representing type IIB string theory as a 12-dimensional theory, F-theory, compactified on a two-torus. This toroidal proof of the S-duality group may also be realized by another embedding: the gauge theory may also be viewed as a long-distance limit of the \(d=6\) \((2,0)\) superconformal field theory compactified on a two-torus; the logic is the same.

You should appreciate that the S-duality is an extremely complicated relationship if you want to construct it or prove it by hand. In fact, it replaces point-like elementary oscillations that are weakly coupled with extended objects such as magnetic monopoles that are strongly coupled. They look like very different physical objects and the proof of the equivalence can't be made in perturbative expansion – because it is not a duality that holds order-by-order in this expansion – but it's still true. But of course, all tests you can fully calculate work: the gauge theory seems to possess the non-trivial S-duality group. In its stringy incarnation, the S-duality may be seen within a second.

Also, Maldacena's holographic duality boils down to the construction of the gauge theory involving D3-branes, too. The low-energy limit of the D3-branes' internal interactions has to be an interacting theory with 16 supercharges – because they aren't being broken by anything – and that has a field content that may be obtained from the counting of open string excitations attached to the D3-branes. You will find out that the theory has to be a gauge theory with the degrees of freedom I enumerated above; the supersymmetries and consistency dictate the interactions uniquely. In the long-distance limit, only the massless open strings i.e. gauge fields and their superpartners matter; closed strings (especially gravity) is decoupled because the energy density per Planck volume is very low in this limit. So we really do have a non-gravitational theory.

On the other hand, the D3-branes in string theory are real objects, lively animals that manifest themselves in many other physical ways. In particular, they have a gravitational field that extends to the transverse dimensions. Much like D0-branes would be particles that would behave as black holes, D3-branes are extended versions of the same objects, extended black holes. We call them black branes or black \(p\)-branes. They are black 3-branes, in this case. Just to be sure, in the previous paragraph, I stated that the gravitational force between the open string interactions may be neglected; but the gravitational field from their substrate – the static D3-branes in which the open strings live – still curves the 10-dimensional spacetime of type IIB string theory.

A funny thing is that if you adopt the full 10-dimensional perspective, the low-energy excitations have another interpretation: they are physical states that are located near the event horizon of the black branes. The relationship between the adjectives "low-energy" and "near-horizon" holds because near the horizon, it's where the excitations that look "very red" from the global viewpoint (of an observer at infinity) may be created in generic processes. That's because of the gravitational red shift, of course.

If you ask which degrees of freedom are kept if you simply consider all low-energy excitations of those 3-branes, you have two methods to answer: you either realize that the 3-branes may be described as D3-branes whose dynamics is governed by interactions of open strings and the low-energy limit of the open strings' interactions is nothing else than the gauge theory; or you may imagine that the D3-branes are actual solutions of a gravitational theory – an extension of general relativity – and low-energy states are the states of all objects that move near the event horizon.

Each of these operations is a valid method to isolate the low-energy states; so the two theories obtained by these methods must be exactly equivalent. That's an elegant proof of the AdS/CFT correspondence, a non-technical, non-constructive proof that avoids almost all mathematics (although one should still add some mathematics in order to show that it really deserves to be called the "proof"). The near-horizon geometry of the black 3-branes is nothing else than \(AdS_5\times S^5\) and gravitational – well, type IIB stringy – phenomena within this spacetime must therefore be exactly described by a four-dimensional gauge theory.

Of course, this successful union of string theory and gauge theory may be extended to other gauge groups, less supersymmetric gauge theories corresponding to less symmetric compactifications of the gravitational side, and even to other dimensions. Lots of objects on both sides of the equivalence may be given new interpretations using the other description, and so on. But the main goal of this text is to describe new field theories and new non-gravitational non-field theories that arise from similar constructions. The most supersymmetric example of the first category is the so-called \((2,0)\) superconformal field theory in 6 dimensions.

M5-branes and their dynamics

In the case of the D3-branes above, we considered objects in string theory in ten dimensions. In the usual weakly coupled approach, these theories are parameterized by the string coupling constant \(g_s\) which is the exponential of the (stringy) dilaton; greetings. The coupling constant is adjustable in the simplest vacua; all values are equally good but the choice isn't a parameter representing inequivalent possibilities. Instead, because the coupling is an exponential of the dilaton and the dilaton is a dynamical field, different values of the coupling constant correspond to different environments that may be achieved in a single theory.

In realistic compactifications, a potential for the dilaton is generated (much like the potential for all other moduli) and string theory picks a preferred value of the string coupling which is at least in principle but – to a large extent – also in practice calculable (much like the detailed shape of the extra dimensions etc.).

However, there exists a vacuum of string/M-theory that has no dilaton-like scalar field that would label inequivalent environments. Of course, it's the 11-dimensional M-theory. The field content of the eleven-dimensional supergravity only includes the graviton, some spin-3/2 gravitino, and spin-1 three-form generalizing electromagnetism. No spin-0 scalar fields here.

That's kind of nice because the theories we may obtain from M-theory in similar ways as the theories obtained from type II or type I or heterotic string theory have an unusual property: they have no adjustable dimensionless coupling constants. This is something we're not used to from the quantum field theory courses taught at schools. In those courses, we first start with a free theory and interactions are added as a voluntary deformation. All these interactions may be chosen to be weak because the coupling constants are adjustable and the free, non-interacting limit is assumed to be OK.

However, for theories obtained from M-theory, we can't turn off the interactions at all! These theories inevitably force their degrees of freedom to interact with a particular vigor that cannot be reduced at all. Because the coupling constants may be measured as the strength of the "quantum processes" – how much the one-loop diagrams where virtual pairs exist for a while are important relatively to the tree-level "classical" processes – we may also say that the theories extracted from M-theory are intrinsically quantum and they have no classical limit.

Are there any?

You bet. As I mentioned in my discussion of 11D SUGRA, the theory has to contain a three-form potential \(C_3\). One may add terms in the Lagrangian where \(C_3\) is integrated over a 3-dimensional world volume in the spacetime. This term generalizes the \(\int \dd x^\mu A_\mu \) coupling of the electromagnetic fields with world lines of charged particles (in the limit in which they're treated as particles with clear world lines, not as fields). And indeed, M-theory does allow such terms; the 3-dimensional world volumes are those of M2-branes, or membranes, objects with 2 spatial and 1 temporal dimensions.

Also, the exterior derivative of the \(C_3\) potential is a four-form \(F_4\) field strength. By using the epsilon symbol in eleven dimensions, this may get mapped to a Hodge-dual seven-form \(F_7\) potential which is locally, in the vacuum, the exterior derivative of a six-form "dual potential" \(C_6\). So M-theory also admits couplings of this \(C_6\) and indeed, the 6-dimensional world volume we integrate over is the world volume of M5-branes, the electromagnetic dual partners of M2-branes.

Just like string theories contain fundamental strings, F1-branes, and lots of heavy D-branes of various dimensions, M-theory contains no strings or 1-branes but it has M2-branes and M5-branes which have different dimensions but are "comparably heavy" as long as their typical mass scale goes.

A nice thing is that just like you may study the long-distance dynamics of D3-branes which led to the very important maximally supersymmetric gauge theory, you may also study the long-distance limit of the dynamics inside M2-branes and M5-branes. Both of them give you some new interesting theories. The theories related to the M2-branes were the subject of the recent "membrane minirevolution"; this was my name for the intense research of some supersymmetric 3-dimensional gauge theories extending the Chern-Simons theory. Some new ways to see the hidden symmetries of these theories were found; the most obvious "clearly new" development of the minirevolution were the ABJM theories extending the long-distance of the membranes to more complicated compactifications. The membrane minirevolution has surprised many people who had thought that such M(ysterious) field theories would never be written in terms of ordinary Lagrangians. They could have been written. People could only discover these very interesting and special Lagrangians once they were forced by string/M-theory to look for them.

When you consider the low-energy limit of the M5-branes, you get a six-dimensional theory: 5 dimensions of space and 1 dimension of time. It is useful to mention how spinors work in 6 dimensions. In 4 dimensions, the minimal spinor is a Weyl spinor (or, equivalently – when it comes to the counting of fields – the Majorana spinor). But there's only one kind: if you include a left-handed Weyl spinor, the theory immediately possesses the Hermitian conjugate right-handed one, too. So you only need to know how many spinors your theory has. For example, the \(\NNN=4\) theory has supercharges that may be organized into 4 Weyl or Majorana spinors.

However, things are a bit different in \(d=6\). Because it is an even number, one still distinguishes left-handed and right-handed Weyl spinors. But in spacetime dimensions of the form \(4k+2\), the left-handed and right-handed spinors are actually not complex conjugates to each other. You may incorporate them independently of each other. The same comment holds for supersymmetries; if you want to accurately describe how the spinors of supersymmetric transform, you must specify how many left-moving and how many right-moving Weyl spinors there are in the list of supercharges.

In ten dimensions, we use the "shortened" terms type I, type IIA, type IIB for \((1,0)=(0,1)\) supersymmetric theories, \((1,1)\) supersymmetric theories, and \((2,0)=(0,2)\) supersymmetric theories, respectively. The permutation of the two labels is immaterial. The type I and type IIB theories are inevitably left-hand-asymmetric i.e. chiral; type IIA is left-right-symmetric i.e. non-chiral, as expected from the fact that it may be produced as a compactification of an 11-dimensional theory.

In six dimensions, there's a similar classification. The \((1,1)\) theories are non-chiral and typically include some gauge fields. On the other hand, the \((2,0)\) theories are chiral. The \((2,0)\) theory we find in the long-distance limit of the M5-branes is non-chiral not only when it comes to the fermions in the field content. Because the labels \((2,0)\) are "very asymmetric" between the first and second digit, the left-right asymmetry actually inevitably gets imprinted to the bosonic spectrum, too. If we're explicit, it's because the theory contains "self-dual field strength fields" i.e. 3-form(s) \(H_3\) generalizing \(F_2\) in Maxwell's theory that however obey \(*H_3=H_3\). Note that this is possible in 6 dimensions but not in 4 dimensions because \((*)^2=+1\) in 6 dimensions but \((*)^2=-1\) in 4 dimensions.

Because the \((2,0)\) theory must allow a generalization of the gauge field whose field strength is however constrained by the self-duality condition, it's hard to write an explicit Lagrangian definition of the theory, at least if we want it to be manifestly Lorentz-symmetric one. It's a part of the unproven lore that this can't be done. However, you must be careful about such widely held beliefs. In particular, the membrane minirevolution has shown that various Lagrangians that would be thought of as impossible are actually totally possible and you never know whether someone will find a clever trick by which this explicit construction may be extended to 6 dimensions.

So the six-dimensional theory can't be constructed as a "quantization" of a classical theory. It's a point that I discussed in less specific contexts in several recent articles about the foundations of quantum mechanics. We see many independent reasons why it's natural that no such "master classical theory" may exist in this case. First, the quantum theory requires the coupling constant to be "one" in some normalization: it can't be adjusted to be close to zero so studying the theory as the deformation of a free theory would be similar to studying \(\pi\) using the \(\pi\to 0\) limit. Second, we have mentioned that the theory contains self-dual fields and it's hard to write a Lagrangian for a potential if you also want its field strength to be self-dual. Third, and it is related, you would have a problem to write renormalizable interactions in a theory in 6 or more dimensions, anyway. A \(\phi^3\) cubic coupling for a scalar would be the "maximum" that would still be renormalizable but it would create instabilities. By denying that there exists a way to represent the full quantum field theory as a quantization of a classical theory (with a polynomial Lagrangian), string/M-theory finds the loophole in all these arguments that a sloppy person could offer as an excuse that such a non-trivial 6-dimensional theory shouldn't exist.

However, this theory still exists as an interacting, non-gravitational theory with all the things you expect from a local quantum field theory. One may define local fields \(\Phi_k(x^\mu)\) and these fields have various correlation functions and may be evolved according to some well-defined Heisenberg equations, and so on. It may be hard or impossible to use the perturbative (and other) techniques we know from the gauge theory but the resulting product – Green's functions etc. – is conceptually identical to the product in the gauge theory. You may be ignorant about methods how to compute these physical answers in the \((2,0)\) theory; but one may actually prove – using the consistency of string theory as a main tool or assumption – that these answers exist and have the same useful properties as similar answers in gauge theory. However, in gauge theory, we may calculate a whole 1-parameter or 2-parameter family of the "collection of Green's functions"; the families are parameterized by the coupling constant (and the axion). In the \((2,0)\) case, there are no such parameters. It's just an isolated theory – one isolated set of Green's functions encoding all the evolution and interactions – without continuously adjustable dimensionless parameters.

Much like the \(\NNN=4\) gauge theory is equivalent to type IIB string theory in \(AdS_5\times S^5\) which we could have derived as the near-horizon geometry of a stack of the D3-branes, the \((2,0)\) theory in six dimensions may be shown to be equivalent to M-theory on \(AdS_7\times S^4\), the near-horizon geometry of a stack of the M5-branes in M-theory. Just to be sure, there is a similar case involving a 3-dimensional Chern-Simons-like theory andd M-theory on \(AdS_4\times S^7\) – note that the labels four and seven got exchanged – which is the near-horizon geometry of a stack of M2-branes in M-theory.

So while the perturbative, weakly coupled methods don't exist for this six-dimensional theory, the holographic AdS/CFT methods work as well as they do for the gauge theory. Also, this six-dimensional theory is as important for Matrix theory, a non-gravitational way to describe some simple enough compactifications of string/M-theory on flat backgrounds, as the gauge theory is. In particular, if you compactify the \((2,0)\) theory on a five-torus (times the real line for time), you get a matrix description for M-theory on a four-torus.

Perturbatively, the \(\NNN=4\) gauge theory with the \(SU(N)\) gauge group seems to have the number of degrees of freedom – independent elementary fields – that scales like \(N^2\). That's because the adjoint representation may be viewed as a square matrix, of course. There are actually different, independent methods to derive this power law, too, in particular a holographic one that is based on the entropy of a dual bulk black hole.

The holographic methods may also be used for the M2-based 3-dimensional theory and the M5-based 6-dimensional theory. They tell you that the number of degrees of freedom in these two theories should scale like \(N^{3/2}\) and \(N^3\) in \(d=3\) and \(d=6\), respectively. The first case, a fractional power, doesn't even produce an integer but it has still been motivated in various ways.

The 6-dimensional case is even more intriguing because the integral exponent does suggest that there could exist a "constructive explanation" – some formulation that uses fields with three "fundamental gauge indices", if you wish. Many authors have tried to shed light on this strange power law. A month ago, Sav Sethi and Travis Maxfield offered a brand new calculation of the "conformal anomaly" (what was interpreted as the number of degrees of freedom) which also produces the right \(N^3\) scaling.

There's still a significant activity addressing this 6-dimensional theory and its less supersymmetric cousins. A few days ago, Elvang, Freedman, Myers, and 3 more colleagues wrote an interesting paper about the a-theorem in six dimensions. You should realize that despite the absence of an old-fashioned, "textbook" Lagrangian classical-based construction of the theory, the amount of knowledge has been growing for more than 15 years. Let me pick my 1998 paper with Ori Ganor as some "relatively early" research of physical effects that occur in this theory.

So the \((2,0)\) theory is conformal and therefore scale-invariant (it is a "fixed point" of the renormalization group) which is why it may occur as the low-energy limit of other physical theories in 6 dimensions; I will mention one momentarily. It has a qualitatively well-understood holographic dual and it appears in a matrix description of M-theory on a four-torus. Some fields, especially the "supersymmetry preserving ones", may be isolated and some of their correlation functions may be calculated purely from SUSY, and so on. The theory has various topological solutions that may be interpreted by various "perspectives" to look at this theory that string/M-theory offers. This six-dimensional theory is also an "ancestor" of the maximally supersymmetric gauge theory; the \(\NNN=4\) gauge theory may be obtained from a compactification of the six-dimensional theory on a two-torus.

There are interesting modifications and projections of this theory, too. For example, there are \((1,0)\) theories in six dimensions which respect an \(E_8\) global symmetry. This global symmetry is inherited from the \(E_8\) gauge symmetry that lives on the domain walls (ends-of-the-world) in M-theory whenever the M5-branes are places on such a boundary. I can't say everything that is interesting about this theory but be sure that there would be lots of other things just to enumerate – and lots of interesting details if I were to fully "teach you" about those things.

One of the broader points is that physics is making progress and finding "conceptually new ways" how to think about old theories, how to calculate their predictions, and how to related previous unrelated physical mechanisms and insights. Quantum field theory is essential in all this research; however, we know that quantum field theory isn't just some mechanical exercise starting from a classical theory and adding interactions to a free limit by perturbative interactions. There are lots of nonperturbative processes and insights that may be obtained without explicit perturbative calculations, too.

Little string theory

I have mentioned that the \((2,0)\) superconformal field theory discussed above was a quantum field theory whose Green's functions are as real as those coming from a gauge theory; they satisfy the same consistency, unitarity, and locality conditions, too. But it's a "fixed point", a scale-invariant theory that may be identified as the "ultimate long-distance limit" of some other theories. Are there any other theories of this kind?

A little string theory is a type of a theory in spacetime that is something in between a quantum field theory in the spacetime; and the full gravitating string theory in the same spacetime. They're not local because we may say that their elementary degrees of freedom or elementary building blocks arise from strings much like in the full string theory; however, an appropriate limit is taken so that the gravitational force between the strings decouples.

This seemingly contradicts the lore that every theory constructed from interacting strings inevitably includes gravity; however, there's actually no congtradiction because while the little string theories contain strings and they are interacting theories, they actually cannot be constructed out of these "elementary strings" by following the usual constructive methods of the full string theory.

Fine, so what is the little string theory? The simplest little string theories carry the same \((2,0)\) supersymmetry in \(d=6\) as the superconformal quantum field theory I was discussing at the beginning. In fact, the long-distance limit of these little string theories (they are parameterized by discrete labels such as the number of 5-branes) produce the superconformal field theory we have already discussed.

But these little string theories are not superconformal or scale-invariant. In fact, they are not local quantum field theories at all. In this sense, they are just a generalization of a quantum field theory in a similar sense as the full string theory is a generalization of a quantum field theory. How can we obtain them?

The most straightforward way to obtain the \((2,0)\) superconformal field theories above were a stack of M5-branes in M-theory. Are there some other objects in string theory that are not M5-branes but that look as M5-branes in the low-energy limit? The answer is Yes. M-theory may be obtained as the strong coupling limit of type IIA string theory. Type IIA string theory also contains 5-branes. But they are not D5-branes which may be found in type IIB string theory; type IIB D5-branes produce \((1,1)\) supersymmetric theories in six dimensions, not \((2,0)\): their world volume is exactly as left-right-symmetric as the type IIB spacetime fails to be. There are also NS5-branes in type IIB string theory which have the same SUSY as the D5-branes, because of S-duality that relates them.

Type IIA string theory only contains D-even-branes, not D5-branes, but it still allows NS5-branes, the electromagnetic duals of fundamental strings. And while type IIA is left-right-symmetric in the spacetime, its NS5-branes are left-right asymmetric; not that there is an anticorrelation between the chirality of the spacetime and the chirality of the NS5-brane world volume.

The dilaton of type IIA string theory has a value that depends on the distance from the NS5-branes; this contrasts with the behavior of D3-branes in type IIB string theory that preserve the constant dilaton (and string coupling) in the whole spacetime. This depends of the dilaton – it goes to infinity near the NS5-branes' core – means that the ultimate low-energy limit of the dynamics of NS5-branes is the same one as it is for M5-branes in M-theory: the new 11th dimension really emerges if you're close enough to the NS5-branes.

On the other hand, one may define a different scaling limit of dynamics inside the type IIA NS5-branes in which the gravity in between the excitations of the NS5-branes is sent to zero; but which is not the ultimate long-distance, scale-invariant limit yet. Such a theory inherits a privileged length scale, the string scale, from the "parent" type IIA string theory. But it doesn't preserve the dilaton or the coupling constant because it's scaled to infinity.

The resulting theory of this limit, the little string theory, has no gravitational force but it has string-like excitations. It is not a local quantum field theory but its low energy limit is a quantum field theory. The theory – which has a "qualitatively higher level of conceptual complexity than the \((2,0)\) superconformal field theory" – also enters Matrix theory; its compactification on a five-torus is the matrix description of M-theory compactified on a five-torus. All the usual limits and dualities between the toroidally compactified string/M-theoretical backgrounds may be deduced from the matrix description, too: these dualities may be reduced to relationships between their non-gravitational matrix descriptions.

The little string theories have various other relationships to quantum field theories and vacua of the full string theory, too. Again, I can't say everything that is known about them and everything that makes them important.

Let me emphasize that none of these theories – neither the new superconformal field theories nor the little string theories – has any adjustable continuous dimensionless parameters. They still have discrete parameters – counting the number of 5-branes in the stack and/or whether or not these 5-branes were positioned at some end-of-the-world boundaries or other singular loci in the parent spacetime. But the absence of the continuously adjustable parameters allows us to say that all these quantum theories are "islands" of a sort.

They're obviously important islands. If you want to study consistent non-gravitational interacting theories in 6 dimensions, these islands may be as important as Hawaii or the Greenland or Polynesia or Africa – it's hard to quantify their importance accurately in this analogy. However, the importance is clearly "finite" and can't go to zero. Hawaii, the Greenland, Polynesia, or Africa inevitably enters many people's lives.

Finally, I want to end up with a more general comment. New exceptional theories that were previously overlooked but that obey all the "quality criteria" that were satisfied by the more well-known theories; and all the new perspectives and "pictures" that allow us to say something or calculate something about these as well as the more ordinary theories are important parts of the genuine progress in theoretical physics and everyone who actually likes theoretical physics must be thrilled by this kind of progress and by the new "concise ways" how some previously impenetrable technical insights may be explained or proved.

There exists a class of people with a very low intelligence, no creativity, no imagination, and no ability to see the "big picture" who are only capable of learning some very limited rules and who are devastated by every new powerful technique or technology that physics learns. These human feces often concentrate around Shmoit-and-Shmolin kind of aggressive sourball crackpot forums. I hope that all readers with IQ above 100 have managed to understand why the text above is enough as a proof of the simple assertion that all these Shmoits-and-Shwolins are just intellecutally worthless dishonest scum.